In the Wikipedia article about kernels, universal algebra section, Mal'cev algebras subsection (https://en.wikipedia.org/wiki/Kernel_(algebra)#Mal.27cev_algebras) appear some so-called Mal'cev algebras (different from the nonassociative ones) which are not actually defined. What should be their rigorous definition? Are they the same as Mal'cev varieties? Is there any good introductory text on universal algebra which touches this topic?
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1AFAIK, a Mal'cev algebra (in Universal Algebra) is some $(A,p)$ where $p:A^3\to A$ that satisfies Mal'cev identities: For all $a,b\in A$, $p(a,b,b)=p(b,b,a)=a$. – Pedro Sánchez Terraf Sep 18 '17 at 00:41
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@PedroSánchezTerraf Yes, but does this imply the existence of a neutral element $0_B$ such that the kernel of a homomorphism $f:A\rightarrow B$ can be equivalently expressed as ${a\in A: f(a)=0_B}$, as is used in Wikipedia? – Jose Brox Sep 18 '17 at 00:53
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The standard definition of a Mal'cev algebra in universal algebra is an algebra $\mathbf{A}$ where there is a term operation $p\in\mathrm{Clo}_3(\mathbf{A})$ staisfying $p(a,b,b)=p(b,b,a)=a$ for all $a,b\in A$.
This does not imply the existence of a neutral element: see for example the variety of quasigroups.
It seems that the Wikipedia page is talking about congruence regular algebras with a one-element subalgebra (but that might be too specific). For more information on congruence regular algebras, see the answer here: Generalization of normal subgroups and ideals.
Eran
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Following citations, I have arrived to two relevant papers:
On the determining of the form of congruences in abstract algebras with equationally definable constant elements, by Slominski: https://eudml.org/doc/213569
The Elementary Character of Two Notions from General Algebra, by Vaught, reviewed here: https://www.jstor.org/stable/2270163